Nuprl Lemma : rev_fill_term_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a1:{Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ u[1]]}].
(rev_fill_term(Gamma;cA;phi;u;a1) ∈ {Gamma.𝕀 ⊢ _:A[(phi)p |⟶ u]})
Proof
Definitions occuring in Statement :
rev_fill_term: rev_fill_term(Gamma;cA;phi;u;a1),
composition-structure: Gamma ⊢ Compositon(A),
partial-term-1: u[1],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-1: 1(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rev-type-line: (A)-,
implies: P ⇒ Q,
all: ∀x:A. B[x],
guard: {T},
squash: ↓T,
prop: ℙ,
true: True,
partial-term-0: u[0],
partial-term-1: u[1],
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-ap-term: (t)s,
interval-rev: 1-(r),
csm-adjoin: (s;u),
interval-0: 0(𝕀),
csm-id: 1(X),
csm-ap: (s)x,
cubical-term-at: u(a),
pi1: fst(t),
pi2: snd(t),
composition-structure: Gamma ⊢ Compositon(A),
rev_fill_term: rev_fill_term(Gamma;cA;phi;u;a1),
cube-context-adjoin: X.A,
context-subset: Gamma, phi,
cubical-type: {X ⊢ _},
cubical-term: {X ⊢ _:A},
and: P ∧ Q,
cubical-type-at: A(a),
I_cube: A(I),
functor-ob: ob(F),
interval-presheaf: 𝕀,
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
so_apply: x[s],
face-type: 𝔽,
face-presheaf: 𝔽,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
bdd-distributive-lattice: BoundedDistributiveLattice,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
interval-rev_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-snd_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
cc-fst_wf_interval,
csm-interval-type,
csm-id-adjoin_wf,
interval-1_wf,
partial-term-1_wf,
constrained-cubical-term-eqcd,
istype-cubical-term,
context-subset_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
thin-context-subset,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cc-fst_wf,
cubical-term_wf,
rev-type-line_wf,
csm_ap_term_fst_adjoin_lemma,
context-subset-map,
csm-adjoin_wf,
csm_id_adjoin_fst_term_lemma,
squash_wf,
true_wf,
csm-ap-id-term,
cubical-type-cumulativity,
rev-type-line-0,
dma-neg-dM0,
partial-term-0_wf,
fill_term_wf,
rev-type-comp_wf,
csm-constrained-cubical-term,
rev-rev-type-line,
cubical-term-equal,
I_cube_pair_redex_lemma,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
DeMorgan-algebra-laws,
dM_wf,
subtype_rel_self,
lattice-point_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
istype-cubical-type-at,
cubical-term-at_wf,
face_lattice_wf,
I_cube_wf,
fset_wf,
nat_wf,
istype-universe,
subset-cubical-type,
context-subset-is-subset,
iff_weakening_equal,
cubical-term-eqcd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
hypothesis,
hypothesisEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
independent_isectElimination,
Error :memTop,
universeIsType,
independent_functionElimination,
dependent_functionElimination,
equalityIstype,
hyp_replacement,
lambdaFormation_alt,
inhabitedIsType,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
cumulativity,
universeEquality,
setElimination,
rename,
functionExtensionality,
productElimination,
dependent_pairEquality_alt,
productEquality,
isectEquality,
dependent_set_memberEquality_alt
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} Compositon(A)].
\mforall{}[u:\{Gamma.\mBbbI{}, (phi)p \mvdash{} \_:A\}]. \mforall{}[a1:\{Gamma \mvdash{} \_:(A)[1(\mBbbI{})][phi |{}\mrightarrow{} u[1]]\}].
(rev\_fill\_term(Gamma;cA;phi;u;a1) \mmember{} \{Gamma.\mBbbI{} \mvdash{} \_:A[(phi)p |{}\mrightarrow{} u]\})
Date html generated:
2020_05_20-PM-04_51_20
Last ObjectModification:
2020_05_02-PM-01_20_34
Theory : cubical!type!theory
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